
was introduced in the balanced OT setting in [1]} for statistical testing, and in [2]}, [3]} for generative learning.
However in those works it was only presented as a heuristic, and observed empirically to yield improved performance in training.
This quantity is proved in [2]} to converge as \(\epsilon \rightarrow \infty \)  to a kernel norm \(\tfrac{1}{2}{- }^2_{^*}\)  with kernel \(k=-. This somehow explains that using Sinkhorn divergences is a way to interpolate between OT and kernel methods.One of the main contributions of~\cite {feydy2018interpolating} and~\cite {sejourne2019sinkhorn} is to prove that under conditions on \) , the Sinkhorn divergence recovers properties of unregularized OT such as positivity, definiteness, and metrization of the weak* convergence.
